sums will set you free

how to teach your child numbers arithmetic
mathematics

subtraction and more counting

how to teach your child number
arithmetic mathematics - subtraction and more countingis part of the series of documents
about fundamental education at abelard.org. These pages are a sub-set
ofsums
will set you free.

This sub-section of abelard.org is designed
to lay out a rational and logical base for teaching arithmetic
and mathematics from basics. I shall not always justify
the methods in this section as I go along, but the methods
are very relevant and purposefully structured. Throughout
this section,many of the yellow links take you to a more advanced, or technical, explanation.

It is vital to understand that there is no fundamental
or logical difference between the symbolism of teaching
English and teaching arithmetic/mathematics. This congruence
becomes part of the learner’s understanding. It
is a deep and dangerous pedagogical error to allow the
learner to imbibe the erroneous concept that mathematics
and English are different “subjects”.

methodology
and examplesOn these pages, you will be given
basic methodology and necessary examples. You will not
be provided with hundreds of examples, those you make
up as you work with the learner, adjusting those examples
according to the person’s problems. Some examples
should be interspersed which are easy for the learner,
in order to reinforce and to give experience of success,
while others should be aimed at specific difficulties.

As you will see, there’s a great deal of stuff
to absorb here, especially if you are three, four or five,
or even eight, nine or ten. Human understanding evolves,
the human absorbs and gradually organises the vast streams
of information coming from those small holes called ears
and eyes.

For example, getting used to and seeing clocks of different
types lays grounds for understanding what the shapes and
numbers mean. Trying to rush these processes leads to
stress and often to confusion - not good.

Letting the child run wild, without any help or guidance,
leaves them struggling to adapt to a civilisation and
culture that has taken thousands of years to develop,
and which is now running in over-drive. It is every bit
as foolish to leave a person in confusion as it is to
feed information too quickly and hammer it in with a mallet.

The purpose of mathematics is to understand patterns
and logic, to help you organise the filing system in your
head. Mathematics is not something esoteric, but there
is rather a lot of it! The sane objective of learning
is not to memorise enormous lists by rote, it is to teach
organised ways of thinking about problems, and where and
how to research for relevant information in the ever growing
data banks of human experience (knowledge).

counting
in the real world

It is widespread common sense that a child learning
to read has their attention drawn to varying sources
of written text. Likewise, it is useful to help a child
to gain numerical fluency as they explore the world.

Keep in mind that this page is to help you in teaching
young children the basics of counting and, here in particular,
subtraction. This section shows various numbers you can
use in the real wore ld to increase the child’s
awareness. There is not the slightest intention or expectation
that the learner is going to gain a comprehensive grasp
of all these various wonders at this point. The purpose
here instead is to generate familiarity and to take opportunities
to engage the person with this modern civilisation.

to count fence posts

to recognise the use of numbers as indices for
bus numbers

to learn methods of counting space by judging
distance using floor tiles in precincts. How far
do you usually step? How far can you step?Tiles in a shopping precinct.
Insert shows measuring tile width/length.

It is important that the child understands that numbers
have many different uses. Counting apples, or trees, of
different sizes has differences from counting inches (or
centimetres), where the aim is to have a fairly constant
size unit.

Subtraction

Subtraction is like addition in reverse,
taking an object, or several objects, from a group
of objects. Take for instance a group of three blocks
on cloths already drawn together.

one two three
3-2
three minus two

Now draw the cloths apart, so separating
one block from the other two

one

two
three

1

2, 3

one

one two

one

two

1

2

one
and two

When the cloths are separated so one
cloth is out of sight, the subtraction is complete.
Two blocks are taken away from three blocks, leaving
one block. Three minus two equals one.

Similarly, we have three blocks and
five blocks.

eight

…

three five

Notice that this can also be looked
at backwards - as eight minus three, by turning
round the pictures, or by going round the table.
And the cloths can be drawn back together again
as an addition. Thus, ideas of reversal are introduced
naturally.

You put on your socks and then your shoes. Will
it work if you put your shoes on first?

You lay out your plate, and then your spoon. If
you lay out your spoon first, does it still work?

In technical language: Are these operations commutative?
Do they commute? The sock example does not, addition
and subtraction do. Often young children like to
learn long and fancy words. In more ordinary language,
are the processes (operations) reversible? It is
also useful to pick up the jargon, as specialists
are often desperately attached to their ‘special’
languages.

Bring the child’s attention to other forms of balance
- balance weighing machines, the see-saw in the local park,
walking across a room and back. Keep in mind that no two
things are the same. The idea of sameness/equals is just a very useful fiction. From time to time, keep the
learner aware that every block in the tower is different,
made out of different wood, by different acts, at different
times. It is very important for a person to fully internalise
this understanding of reality. Each duck on the pond is
different, each step taken is different. As I regularly
point out, the world is changing all
the time. Even the blocks are under continual change,
as the child breathes and grows.

The purpose of this section is to encourage and develop
an understanding of the formation of collections. It will
be soon needed in the next stage (multiplication). If
you do not have logic blocks (yet?), onions and stones
and insects and flowers and boxes of detergent will serve.

Logic blocks, also known as attribute blocks, are
made as a set of plastic shapes. This is a four attribute
set - shape, size, thickness and colour. The blocks
can be sorted into groups of a specific attribute
or several attributes.

Sorted according to thickness (and by shape)

Sorted by size (and by shape)

Sorted by colour

sorted by shape (and size and thickness)

Logic blocks can also be laid out around an infant
to provide bright stimulus of shapes. Logic blocks
are also handy for teething.

abelard.org
maths educational counter

[This counter functions with javascript,
you need to ensure that javascript is enabled for the
counter to work.]

On this page is a more concise version of the Brilliant
educational maths counter. The full version with more
detailed instructions, go to the
introduction page.

So, to practise doing subtractions,

Reset Counter Value to 2;

Change Step to 1;

Switch Direction to Decreasing;

Now click on the Manual Step button.

The counter counts down (decreasing): 2, 1, 0, -1, -2 and so on.
Then you could change the Step size to 3! Encourage the learner to try
many starting values (Counter Value)
and numbers to subtract (Decreasing Steps).

[This
counter functions with javascript, you need to ensure
that javascript is enabled for the counter to work.]

Is the counter
Manual or Automatic? :

You have done
manual steps since the last reset

Decimal Places
[between 0 and 5]: the counter is displayed up to
decimal places

Estimating distance by pacing:
remember a child’s pace length will change as
they grow . Bring this to their attention, have a height-measuring
point - a door jamb is one useful position. Understanding
change and movement is an important basic concept.

Analogue and digital
A typical analogue device is a clock in which the hands
move continuously around the face. Such a clock is capable
of indicating every possible time of day. In contrast,
a digital clock is capable of representing only a finite
number of times (every tenth of a second, for example).

An arrow flying through the air is continuous movement.
Counting the number of arrows you have is digital. For
thousands of years, through using numbers to describe
the flight of the arrow, humans have become confused
by these two different uses of numbers.

Reversing an operation is often
loosely referred to as ‘doing the opposite’.
For example, the opposite of riding your bicycle from
Oxford to London, in clement weather, when you’re
fit and fresh, is riding back at night, in pouring rain,
after a hard day’s partying.

Logic or attribute blocks are available from several sources. A reliable source is from amazon.com or amazon.co.uk.
An attribute blocks class set, also called a giant or jumbo set, similar to the set illustrated
above, costs $25.95 [at amazon.com, as at 05/2013] or £30.99 [amazon.co.uk, as at 05/2013]. Its shipping weight is 6lb/4kg.

sums will
set you free includes the series
of documents about economics and money at abelard.org.